The paper introduces a new method for performing nonlinear Principal Component Analysis (PCA) using kernel functions. By representing dot products in feature spaces, the method efficiently computes principal components in high-dimensional spaces related to input space through a nonlinear map. The authors derive the method, discuss its application to other techniques that can be made nonlinear with the kernel approach, and present initial experimental results on nonlinear feature extraction for pattern recognition. The paper covers the derivation of the method, its properties, and comparisons with other nonlinear PCA methods. It also discusses the computational complexity and interpretability of the method, and provides examples of its application in object recognition and character recognition tasks. The authors conclude by highlighting the potential of the kernel-based approach in constructing nonlinear variants of various algorithms and its advantages over traditional methods.The paper introduces a new method for performing nonlinear Principal Component Analysis (PCA) using kernel functions. By representing dot products in feature spaces, the method efficiently computes principal components in high-dimensional spaces related to input space through a nonlinear map. The authors derive the method, discuss its application to other techniques that can be made nonlinear with the kernel approach, and present initial experimental results on nonlinear feature extraction for pattern recognition. The paper covers the derivation of the method, its properties, and comparisons with other nonlinear PCA methods. It also discusses the computational complexity and interpretability of the method, and provides examples of its application in object recognition and character recognition tasks. The authors conclude by highlighting the potential of the kernel-based approach in constructing nonlinear variants of various algorithms and its advantages over traditional methods.